--- a/src/HOL/Isar_examples/KnasterTarski.thy Mon Jun 22 22:51:08 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,111 +0,0 @@
-(* Title: HOL/Isar_examples/KnasterTarski.thy
- Author: Markus Wenzel, TU Muenchen
-
-Typical textbook proof example.
-*)
-
-header {* Textbook-style reasoning: the Knaster-Tarski Theorem *}
-
-theory KnasterTarski
-imports Main Lattice_Syntax
-begin
-
-
-subsection {* Prose version *}
-
-text {*
- According to the textbook \cite[pages 93--94]{davey-priestley}, the
- Knaster-Tarski fixpoint theorem is as follows.\footnote{We have
- dualized the argument, and tuned the notation a little bit.}
-
- \textbf{The Knaster-Tarski Fixpoint Theorem.} Let @{text L} be a
- complete lattice and @{text "f: L \<rightarrow> L"} an order-preserving map.
- Then @{text "\<Sqinter>{x \<in> L | f(x) \<le> x}"} is a fixpoint of @{text f}.
-
- \textbf{Proof.} Let @{text "H = {x \<in> L | f(x) \<le> x}"} and @{text "a =
- \<Sqinter>H"}. For all @{text "x \<in> H"} we have @{text "a \<le> x"}, so @{text
- "f(a) \<le> f(x) \<le> x"}. Thus @{text "f(a)"} is a lower bound of @{text
- H}, whence @{text "f(a) \<le> a"}. We now use this inequality to prove
- the reverse one (!) and thereby complete the proof that @{text a} is
- a fixpoint. Since @{text f} is order-preserving, @{text "f(f(a)) \<le>
- f(a)"}. This says @{text "f(a) \<in> H"}, so @{text "a \<le> f(a)"}.
-*}
-
-
-subsection {* Formal versions *}
-
-text {*
- The Isar proof below closely follows the original presentation.
- Virtually all of the prose narration has been rephrased in terms of
- formal Isar language elements. Just as many textbook-style proofs,
- there is a strong bias towards forward proof, and several bends in
- the course of reasoning.
-*}
-
-theorem Knaster_Tarski:
- fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
- assumes "mono f"
- shows "\<exists>a. f a = a"
-proof
- let ?H = "{u. f u \<le> u}"
- let ?a = "\<Sqinter>?H"
- show "f ?a = ?a"
- proof -
- {
- fix x
- assume "x \<in> ?H"
- then have "?a \<le> x" by (rule Inf_lower)
- with `mono f` have "f ?a \<le> f x" ..
- also from `x \<in> ?H` have "\<dots> \<le> x" ..
- finally have "f ?a \<le> x" .
- }
- then have "f ?a \<le> ?a" by (rule Inf_greatest)
- {
- also presume "\<dots> \<le> f ?a"
- finally (order_antisym) show ?thesis .
- }
- from `mono f` and `f ?a \<le> ?a` have "f (f ?a) \<le> f ?a" ..
- then have "f ?a \<in> ?H" ..
- then show "?a \<le> f ?a" by (rule Inf_lower)
- qed
-qed
-
-text {*
- Above we have used several advanced Isar language elements, such as
- explicit block structure and weak assumptions. Thus we have
- mimicked the particular way of reasoning of the original text.
-
- In the subsequent version the order of reasoning is changed to
- achieve structured top-down decomposition of the problem at the
- outer level, while only the inner steps of reasoning are done in a
- forward manner. We are certainly more at ease here, requiring only
- the most basic features of the Isar language.
-*}
-
-theorem Knaster_Tarski':
- fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
- assumes "mono f"
- shows "\<exists>a. f a = a"
-proof
- let ?H = "{u. f u \<le> u}"
- let ?a = "\<Sqinter>?H"
- show "f ?a = ?a"
- proof (rule order_antisym)
- show "f ?a \<le> ?a"
- proof (rule Inf_greatest)
- fix x
- assume "x \<in> ?H"
- then have "?a \<le> x" by (rule Inf_lower)
- with `mono f` have "f ?a \<le> f x" ..
- also from `x \<in> ?H` have "\<dots> \<le> x" ..
- finally show "f ?a \<le> x" .
- qed
- show "?a \<le> f ?a"
- proof (rule Inf_lower)
- from `mono f` and `f ?a \<le> ?a` have "f (f ?a) \<le> f ?a" ..
- then show "f ?a \<in> ?H" ..
- qed
- qed
-qed
-
-end